The system of conservation laws (REF ) can be written in the form \(Wp=S\) , where, by (REF ), \(W\)  is the generalised Laplacian matrix of the graph \({\cal G}\) , depending of \(D_{ij}\)  and \(L_{ij}\) , and \(W=W^T\) , [1]}. The vector \(S\)  describes the sources and the sinks, obeys condition (REF ), and \(p\)  is the vector of pressures. If \(i\ne j\)  and \((i,j) \in E\) , then \(w_{ij}<0\) . If \(i\ne j\)  and \((i,j) \notin E\) , then \(w_{ij}=0\) . As \(w_{ii}>0\) , the matrix \(W\)  is diagonally dominated.
As the graph \({\cal G}\)  has one component, then \(\hbox{rank}(W)=N-1\) , [2]}.
As \(W{\bf 1}=0\) , the eigenvector \({\bf 1}\)  of \(W\)  corresponds to the zero eigenvalue with multiplicity 1.
Then, the system of equations (REF ) can be solved for all the pressures, up to an additive constant, and the fluxes are well determined by (REF ).
